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\begin{table}[!tbp]
\caption{Simulation results: Exponential outcome models\label{tb_Exponential_2}} 
{\centering
\begin{tabular}{lrrcrcrrcrrcrrcrcrr}
\hline
\multicolumn{1}{l}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}\tabularnewline
\cline{2-3} \cline{7-8} \cline{13-14} \cline{18-19}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Exponential outcome model 1: correct PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$-14.23$&$17.95$&&$$&&$-14.34$&$ 15.16$&&$$&$$&&$ 14.79$&$19.57$&&$$&&$ 14.41$&$15.45$\tabularnewline
~~\textbf{nDBW DR}&$ -3.58$&$10.77$&&$$&&$ -0.94$&$  5.06$&&$$&$$&&$ -1.10$&$10.49$&&$$&&$ -0.32$&$ 4.81$\tabularnewline
~~MLE DR&$ -2.44$&$13.31$&&$$&&$ -0.58$&$  6.69$&&$$&$$&&$ -0.58$&$12.93$&&$$&&$ -0.22$&$ 5.52$\tabularnewline
~~CBPS DR&$ -2.64$&$11.91$&&$$&&$ -0.75$&$  5.85$&&$$&$$&&$ -0.67$&$11.60$&&$$&&$ -0.22$&$ 5.21$\tabularnewline
~~Calibrated weighting DR&$ -2.94$&$10.97$&&$$&&$ -0.86$&$  5.31$&&$$&$$&&$ -0.88$&$10.69$&&$$&&$ -0.28$&$ 4.82$\tabularnewline
~~Entropy balancing DR&$ -5.56$&$11.72$&&$$&&$ -4.10$&$  6.35$&&$$&$$&&$ -2.60$&$10.61$&&$$&&$ -2.05$&$ 4.96$\tabularnewline
~~True propensity score DR~~&$ -2.67$&$13.96$&&$$&&$ -0.55$&$  6.98$&&$$&$$&&$ -0.85$&$13.00$&&$$&&$ -0.31$&$ 5.90$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Exponential outcome model 1: misspecified PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$-14.52$&$18.28$&&$$&&$-14.28$&$ 15.09$&&$$&$$&&$ 14.28$&$18.98$&&$$&&$ 14.47$&$15.48$\tabularnewline
~~\textbf{nDBW DR}&$ -5.13$&$11.96$&&$$&&$ -2.63$&$  5.88$&&$$&$$&&$  2.05$&$11.98$&&$$&&$  2.78$&$ 5.88$\tabularnewline
~~MLE DR&$  0.81$&$61.38$&&$$&&$ 45.24$&$601.02$&&$$&$$&&$  3.69$&$14.97$&&$$&&$  4.11$&$ 7.91$\tabularnewline
~~CBPS DR/BRDR&$ -5.84$&$13.06$&&$$&&$ -2.35$&$  6.95$&&$$&$$&&$  4.23$&$13.75$&&$$&&$  3.96$&$ 6.94$\tabularnewline
~~Calibrated weighting DR&$ -5.15$&$12.06$&&$$&&$ -2.62$&$  6.04$&&$$&$$&&$  2.61$&$12.17$&&$$&&$  3.17$&$ 6.08$\tabularnewline
~~Entropy balancing DR&$ -7.48$&$13.07$&&$$&&$ -5.40$&$  7.52$&&$$&$$&&$  0.83$&$11.72$&&$$&&$  1.23$&$ 5.15$\tabularnewline
~~True propensity score DR~~&$ -2.69$&$13.86$&&$$&&$ -0.47$&$  7.26$&&$$&$$&&$ -0.99$&$13.08$&&$$&&$ -0.34$&$ 5.91$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Exponential outcome model 2: correct PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ 14.03$&$18.73$&&$$&&$ 14.34$&$ 15.41$&&$$&$$&&$-14.39$&$18.03$&&$$&&$-14.31$&$15.13$\tabularnewline
~~\textbf{nDBW DR}&$ -2.85$&$11.43$&&$$&&$ -0.77$&$  5.09$&&$$&$$&&$ -5.32$&$12.05$&&$$&&$ -1.41$&$ 5.48$\tabularnewline
~~MLE DR&$ -1.91$&$14.22$&&$$&&$ -0.20$&$  6.55$&&$$&$$&&$ -2.77$&$17.88$&&$$&&$ -0.33$&$ 9.64$\tabularnewline
~~CBPS DR&$ -2.35$&$12.78$&&$$&&$ -0.42$&$  6.00$&&$$&$$&&$ -3.17$&$13.59$&&$$&&$ -0.69$&$ 7.28$\tabularnewline
~~Calibrated weighting DR&$ -2.67$&$11.79$&&$$&&$ -0.60$&$  5.40$&&$$&$$&&$ -3.78$&$12.00$&&$$&&$ -1.00$&$ 5.95$\tabularnewline
~~Entropy balancing DR&$ -7.04$&$13.28$&&$$&&$ -5.60$&$  7.53$&&$$&$$&&$ -8.70$&$13.92$&&$$&&$ -6.93$&$ 8.60$\tabularnewline
~~True propensity score DR~~&$ -2.06$&$14.14$&&$$&&$ -0.31$&$  6.79$&&$$&$$&&$ -3.27$&$19.39$&&$$&&$ -0.55$&$10.54$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Exponential outcome model 2: misspecified PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ 14.67$&$19.55$&&$$&&$ 14.42$&$ 15.51$&&$$&$$&&$-14.46$&$18.11$&&$$&&$-14.24$&$15.12$\tabularnewline
~~\textbf{nDBW DR}&$  7.97$&$15.57$&&$$&&$  9.73$&$ 11.41$&&$$&$$&&$-11.74$&$15.76$&&$$&&$ -8.50$&$ 9.80$\tabularnewline
~~MLE DR&$ 20.04$&$59.54$&&$$&&$ 49.55$&$355.21$&&$$&$$&&$-13.17$&$18.47$&&$$&&$-12.22$&$13.54$\tabularnewline
~~CBPS DR/BRDR&$ 10.71$&$18.34$&&$$&&$ 13.09$&$ 14.81$&&$$&$$&&$-13.05$&$17.38$&&$$&&$-12.94$&$14.02$\tabularnewline
~~Calibrated weighting DR&$  8.62$&$16.01$&&$$&&$ 10.26$&$ 11.93$&&$$&$$&&$-11.35$&$15.56$&&$$&&$ -9.81$&$11.04$\tabularnewline
~~Entropy balancing DR&$  4.84$&$14.05$&&$$&&$  6.36$&$  8.68$&&$$&$$&&$-16.36$&$19.28$&&$$&&$-15.34$&$16.07$\tabularnewline
~~True propensity score DR~~&$ -1.59$&$14.40$&&$$&&$ -0.43$&$  6.71$&&$$&$$&&$ -3.20$&$19.84$&&$$&&$ -0.49$&$10.52$\tabularnewline
\hline
\end{tabular}}
\parbox{0.99\textwidth}
	{Notes: This simulation compares the performance of various methods 
	for estimating propensity scores and (inverse probability) weights 
	by investigating combinations of two versions of the true outcome model 
	(Exponential~1 and 2)
	and two versions of coefficients for the propensity score model (type~A and B)
	with the two different numbers of observations ($n = 200$ and $n = 1000$).
	For each estimation method, I use two propensity score model specifications 
	(correct and misspecified) and report the bias and RMSE for each in the table.}\end{table}
